Introduction to Accelerator Physics
2011 Mexican Particle Accelerator School
Lecture 4/7: Stability, FODO Cells, More Lattice
Functions, Emittance, Chromaticity (and
maybe Dispersion)
Todd Satogata (Jefferson Lab)
[email protected]
http://www.toddsatogata.net/2011-MePAS
Friday, September 30, 2011
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Review
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Transport Matrix Stability Criteria
  For long systems (rings) we want
  If 2x2 M has eigenvectors
stable as
and eigenvalues
  M is also unimodular (det M=1) so
  For
to remain bounded, must be real
:
with complex
  We can always transform M into diagonal form with the
eigenvalues on the diagonal (since det M=1); this does not
change the trace of the matrix
  The stability requirement for these types of matrices is then
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Periodic Example: FODO Cell Phase Advance
cell
  Select periodicity between centers of focusing quads
  A natural periodicity if we want to calculate maximum β(s)
 
only has real solutions (stability) if
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Periodic Example: FODO Cell Beta Max/Min
cell
  What is ?
  A natural periodicity if we want to calculate maximum β(s)
  Follow a similar strategy reversing F/D quadrupoles to find
the minimum β(s) within a FODO cell (center of D quad)
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FODO Betatron Functions vs Phase Advance
Generally want beta_max/L small
Strong focusing
Smaller magnets
Less expensive accelerator
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FODO Beta Function, Betatron Motion
Betatron motion
trajectory
design trajectory
  This is a picture of a FODO lattice, showing contours of
since the particle motion goes like
  This also shows a particle oscillating through the lattice
  Note that
provides an “envelope” for particle oscillations
• 
is sometimes called the envelope function for the lattice
  Min beta is at defocusing quads, max beta is at focusing quads
  6.5 periodic FODO cells per betatron oscillation
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Example: RHIC FODO Lattice
low-beta insertion
low-beta insertion
Horizontal
Vertical
matching
FODO
  1/6 of one of two RHIC synchrotron rings, injection lattice
  FODO cell length is about L=30m
  Phase advance per FODO cell is about

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General Non-Periodic Transport Matrix
  We can parameterize a general non-periodic transport
matrix from s1 to s2 using the lattice parameters
  This does not have a pretty form like the periodic matrix
However both can be expressed as
where the C and S terms are cosine-like and sine-like; the
second row is the s-derivative of the first row!
The most common use of this matrix is the m12 term:
Effect of angle kick
on downstream position
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(Deriving the Non-Periodic Transport Matrix)
Calculate A, B in terms of initial conditions
and
Substitute (A,B) and put into matrix form:
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Propagating Lattice Parameters
  If I have
and I have the transport matrix
that transports particles from
, how do I find the
new lattice parameters
?
The J(s) matrices at s1, s2 are related by
Then expand, using det M=1
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========== What’s the Ellipse? ==========
  Area of an ellipse that envelops a given percentage of the
beam particles in phase space is related to the emittance
We can express this in terms of our lattice functions!
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Invariants and Ellipses
  We assumed A was constant, an invariant of the motion
(A4)
A can be expressed in terms of initial coordinates to find
This is known as the Courant-Snyder invariant: for all s,
Similar to total energy of a simple harmonic oscillator
looks like an elliptical area in
phase space
Our matrices look like scaled rotations (ellipses) in phase space
s=s0
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s=s0+C
s=s0+2C
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s=s0+3C
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Emittance
  The area of the ellipse inscribed by any given particle in
phase space as it travels through our accelerator is called
the emittance : it is constant (A4) and given by
Emittance is often quoted as the area of the ellipse that would
contain a certain fraction of all (Gaussian) beam particles
e.g. RMS emittance contains 39% of 2D beam particles
Related to RMS beam size
Yes, this RMS beam size depends on s!
RMS emittance convention is fairly standard
for electron rings, with units of mm-mrad
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Adiabatic Damping and Normalized Emittance
  But assumption
 
 
 
 
 
 
(A4)
is violated when we accelerate!
When we accelerate, invariant emittance is not invariant!
We are defining areas in
phase space
The definition of x doesn’t change as we accelerate
But
does since
changes!
scales with relativistic beta, gamma:
This has the effect of compressing x’ phase space by
  Normalized emittance is the invariant in this case
unnormalized emittance goes down as we accelerate
This is called adiabatic damping, important in, e.g., linacs
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Phase Space Ellipse Geography
y2
y1
x1 x2
  Now we can figure out some things from a phase space
ellipse at a given s coordinate:
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Rings and Tunes
  A synchrotron is by definition a periodic focusing system
  It is very likely made up of many smaller periodic regions too
  We can write down a periodic one-turn matrix as before
  Recall that we defined tune as the total betatron phase
advance in one revolution around a ring divided by
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Tunes
  There are horizontal and vertical tunes
  turn by turn oscillation frequency
  Tunes are a direct indication of the amount of focusing in
an accelerator
  Higher tune implies tighter focusing, lower
  Tunes are a critical parameter for accelerator performance
  Linear stability depends greatly on phase advance
  Resonant instabilities can occur when
  Often adjusted by changing groups of quadrupoles
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Chromaticity
  Just like bending depended on momentum (dispersion),
focusing (and thus tunes) depend on momentum
  The variation of tunes with is called chromaticity
  Insert a momentum perturbation is like adding a small extra
focusing to our one-turn matrix that depends on the
unperturbed focusing K0
  This looks painful, but remember the trace is related to the
new tune
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Chromaticity Continued
  These last two terms must be equal, which gives
Integrate around the ring to find the total tune change
Natural Chromaticity is defined as
The tune Q invariably has some spread from momentum spread
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Homework
  Design a circular synchrotron made of 20 identical FODO cells,
with bending dipoles in place of the drifts for 500 MeV electrons
  What is the bend angle of each dipole?
•  For 1.5 T maximum dipole field, how long is each dipole?
•  How long is each FODO cell assuming the quads are thin quads?
  Assume a reasonable FODO phase advance per cell
•  Treat the dipoles as drifts for the following analysis
  Calculate
•  The minimum and maximum beta of each FODO cell
•  The tunes QX and QY
•  The natural chromaticities ξX and ξY (hint: the integral on p.20
becomes a sum for thin quadrupoles)
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Dispersion
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